4. DYNAMICAL ANALYSIS OF KINEMATIC DATA

This Section gives a brief overview of how the gravitational potential
of an elliptical galaxy can be inferred from stellar-kinematic
data. These data may include PN velocities, which individually sample
the stellar line-of-sight velocity distributions L(v)
(hereafter
LOSVD) at their positions in the galaxy image, as well as ALS
spectroscopy measurements of the first few moments of the LOSVD, at a
set of 1D or 2D binned positions.

As is well known, velocity dispersion profile measurements (and
streaming velocities, if the galaxy rotates) do not suffice to
determine the distribution of mass with radius, due to the degeneracy
with orbital anisotropy. With very extended measurements a constant
M / L model can be ruled out (e.g.,
[18]),
but the detailed M(r)
still remains undetermined. Thus mass determination in elliptical
galaxies always involves determining the orbital structure at the same
time, and requires a lot of data. LOSVDs from absorption line profile
shapes provide additional data with which the degeneracy between orbit
structure and mass can largely be broken. Simple spherical models are
useful to illustrate this
[24]:
at large radii, radial orbits are seen
side-on, resulting in a peaked LOSVD (positive Gauss-Hermite parameter
h4), while tangential orbits lead to a flat-topped or
double-humped LOSVD (h4 < 0). Similar
considerations can be made for edge-on or face-on disks
[25]
and spheroidal systems
[26].

One may think of the LOSVDs constraining the anisotropy, after which
the Jeans equations can be used to determine the mass distribution.
However, the gravitational potential influences not only the widths,
but also the shapes of the LOSVDs (see illustrations in
[24]).
Furthermore, eccentric orbits visit a range of galactic radii and may
therefore broaden a LOSVD near their pericentres as well as leading to
outer peaked profiles. Thus, in practice, the dynamical modelling to
determine the orbital anisotropy and gravitational potential must be
done globally, and is typically done in the following steps:

(2) write down a composite distribution function (DF) f =
kakfk, where the
fk can be orbits, or DF components such as
fk(E, L2), with free
ak;

(3) project the fk to observed space,
pjk =
Kjfkd,
where Kj is the projection operator for observable
Pj, and
denotes the line-of-sight
coordinate and the velocities;

(4) fit the data Pj =
kakpjk for all observables
Pj simultaneously, minimizing a
2 or
negative likelihood, and including regularization to avoid spurious large
fluctuations in the solution. This determines the ak,
i.e, the best DF f, given
, which must be f
> 0 everywhere;

(5) vary , go back to
(1), and determine confidence limits
on the parameters of .

If the mass distribution and gravitational potential are known from
analysis of, e.g., X-ray data, step (5) can be omitted. Because this
eliminates the degeneracy with orbital anisotropy, considerably fewer
data are then needed.

Such a scheme has been employed regularly to model spherical and
axisymmetric galaxies [e.g.,
4,
27],
using mostly orbits ("Schwarzschild's method") and, more rarely,
distribution function
components. Discrete velocity data were modelled in [e.g.,
5,
17].
The modelling techniques used to constrain black hole masses from
nuclear kinematics and dark halo parameters from extended kinematics
are very similar.

Line-profile shape parameter measurements are now available for many
nearby ellipticals, but those reaching to
~ 2Re are still scarce [e.g.,
28,
4].
Modelling of the outer mass profiles of
ellipticals from such data has been done for some two dozen round
galaxies in the spherical approximation, and for a few cases using
axisymmetric three-integral models.

From recent SAURON integral field measurements it has become clear
that the kinematics of many elliptical galaxies show features
imcompatible with axisymmetry
[16].
This has motivated Verolme et al. to extend Schwarzschild's
method to triaxial models
[29], a
development that has only recently become possible with the increase
of available computing power. In parallel, adaptive N-body codes are
being developed that use the algorithm of Syer & Tremaine for
training N-body systems to adapt to a specified set of data
constraints
[30].
These latter models have the additional advantage of
simultaneously providing a check for the model's dynamical
stability. One application has been to the rotating Galactic bar; work
currently in progress is on axisymmetric and rotating triaxial galaxies.